I'm not certain about the Corvette but in the mid '80s GM was using a gadget called a odometer stepper motor to drive a mechanical odometer.
That may be the only thing wrong.
Before disassembling the instrument cluster I would get the shop manual from www.alldatadiy.com
Once you locate the stepper, disconnect it and test it with an ohmmeter to see if burned out or shorted.

Vic,you are right, the 85 corvette is the same way. the killer on the corvettes is the disassembly of the instrument cluster, to the OP-be very careful with the cluster disassembly, there are 3 full sized circuit boards stacked and connected together and a full size flexable plastic deal with gold/copper paint to connect all circuits and lights together. PS, change ALL THE LIGHT BULBS in there while you are at it, you will be glad you did.

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